Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-18: Find the specified term of the arithmetic sequence that has the two given terms. $$ a_{1} ; \quad a_{8}=47, \quad a_{9}=53 $$

Step-by-Step Solution

Answer

The first term, \( a_1 \), is 5.

Step 1: Understand the Problem

We need to find the first term \( a_1 \) of an arithmetic sequence given two terms: \( a_8 = 47 \) and \( a_9 = 53 \).

Step 2: Write the General Term Formula

The general formula for the \( n \)-th term of an arithmetic sequence is \( a_n = a_1 + (n-1) imes d \), where \( d \) is the common difference.

Step 3: Express Known Terms in Formula

Using the formula, express \( a_8 \) and \( a_9 \):\[a_8 = a_1 + 7d = 47 \]\[a_9 = a_1 + 8d = 53 \]

Step 4: Find the Common Difference \( d \)

Subtract the equation of \( a_8 \) from \( a_9 \):\[\begin{align*}a_9 - a_8 &= (a_1 + 8d) - (a_1 + 7d) \53 - 47 &= d \d &= 6\end{align*}\]The common difference \( d \) is 6.

Step 5: Find the First Term \( a_1 \)

Substitute \( d = 6 \) back into the equation for \( a_8 \):\[\begin{align*}a_8 &= a_1 + 7d \47 &= a_1 + 7 \times 6 \47 &= a_1 + 42 \a_1 &= 47 - 42 \a_1 &= 5\end{align*}\]The first term \( a_1 \) is 5.

Key Concepts

Common Difference

The common difference in an arithmetic sequence is a crucial concept that determines the distance between consecutive terms. In simple terms, it is the fixed amount that each term increases or decreases by from one to the next. To understand it better, imagine hiking on a steady hill where every step you take is exactly the same height or slope upward. This consistent change is represented by the common difference.

In the given exercise, we've been provided with two terms of the sequence: the eighth term (\(a_8 = 47\)) and the ninth term (\(a_9 = 53\)). To find the common difference, we look at how much the sequence increases from one term to the next. Here, it increases by 6, calculated as follows:

• Subtract the 8th term from the 9th term: \(a_9 - a_8 = 53 - 47 = 6\)
• Thus, the common difference \(d = 6\).

This common difference plays a significant role, helping us find other terms in the sequence and understanding the pattern of progression.

General Term Formula

The general term formula of an arithmetic sequence is a powerful tool that allows us to find any term in the sequence without listing all previous terms. Given as \(a_n = a_1 + (n-1) \times d\), this formula uniquely defines each term based on its position (n) in the sequence.

Here's how each component plays a role:

• \(a_n\): This is the nth term we want to find.
• \(a_1\): This is the first term of the sequence.
• \(n\): This represents the position of the term within the sequence.
• \(d\): Our common difference, indicating the consistent interval between terms.

Using the example from our exercise, the general term for the sequence is established, and we use known values \(a_8 = 47\) and \(a_9 = 53\) to solve for \(d\) and determine other required terms. Once \(d\) is known, it becomes straightforward to find the first term or any other term in the sequence by substituting back into the formula.

Arithmetic Progression

An arithmetic progression (AP) solves the mystery of sequences by defining patterns through regular steps in its terms. Simply put, an AP is a number arrangement where successive numbers differ by a constant, called the common difference, that we explored earlier. Arithmetic sequences are everywhere, such as in schedules where events occur at fixed intervals.

In our specific example, knowing two terms \(a_8 = 47\) and \(a_9 = 53\) allows us to dive deep into the structure of this arithmetic progression. Each number follows the one before it by the common difference, highlighting the intricacy of mathematical sequence patterns.

The power of understanding APs lies in their predictability and simplicity. Once we decipher the sequence's formula, we can predict future numbers or find missed ones. This fundamental concept not only aids in solving problems like those in the exercise but also enhances our grasp of patterns in various real-world contexts.